NUMBER SYSTEMS
WHAT IS NUMBER SYSTEM?
Like many words and phrases, the phrase “number system” has more than one meaning.
- A number system is nothing but a way to represent numbers.
- A collection of things (usually called numbers) together with operations on those numbers and the properties that the operations satisfy.
- A system for representing (that is expressing or writing) numbers of a certain type.
- a collection of numbers together with operations, properties of the operations, and a system of representing these numbers.
TYPES OF NUMBER SYSTEMS
AS PER REQUIREMENTS THERE ARE VARIOUS TYPES OF NUMBER SYSTEMS.
- NON POSITIONAL NUMBER SYSTEM
- POSITIONAL NUMBER SYSTEM
- DECIMAL NUMBER SYSTEM
- BINARY NUMBER SYSTEM
- OCTAL NUMBER SYSTEM
- HEXADECIMAL NUMBER SYSTEM
Non POSITIONAL NUMBER SYSTEMS
- Use only a few symbols called digits
- These symbols represent different values depending on the position occupy in the number
POSITIONAL NUMBER SYSTEMS
- Use only a few symbols called digits
- These symbols represent different values depending on the position occupy in the number
- The value of each digit is determined by :
- The digit itself
- The position of the digit in the number
- The base of the number system
( base = total number of digits in the number system )
- The maximum value of a single digit is always equal to one less the value of the base
Example of DECIMAL NUMBER SYSTEMS
236810 = ( 2 x 103 ) + ( 3 x 102 ) + ( 6 x 101 ) + ( 8 x 100 )
= 2000 + 300 + 60 + 8
BINARY NUMBER SYSTEMS
- A positional number system
- Has only 2 symbols or digits ( 0, 1 ). Hence, its base = 2
- The maximum value of a single digit is 1 ( one less than the value of the base)
- Each position of a digit represents a specific power of the base (2)
- We use this number system in our computers
Example of BINARY NUMBER SYSTEMS
101012 = (1 x 24 ) + ( 0 x 23 ) + ( 1 x 22 ) + ( 0 x 21 ) + ( 1x 20 )
= 16 + 0 + 4 + 0 + 1
= 2110
BIT
- Bit stands for binary digit
- A bit in computer terminology means either a 0 or a 1
- A binary number consisting of n bits is called an n-bit number
OCTAL NUMBER SYSTEMS
- A positional number system
- Has only 8 symbols or digits ( 0, 1, 2, 3, 4, 5, 6, 7 ). Hence, its base = 2
- The maximum value of a single digit is 7 ( one less than the value of the base)
- Each position of a digit represents a specific power of the base (8)
Example of OCTAL NUMBER SYSTEMS
Since there are only 8 digits, 3 bits ( 23 = 8 ) are sufficient represent any octal number in binary
Example
20578 = ( 2 x 83 ) + ( 0 x 82 ) + ( 5 x 81 ) + ( 7 x 80 )
= 1024 + 0 + 40 + 7
= 107110
HEXADECIMAL NUMBER SYSTEMS
- A positional number system
- Has total 16 symbols or digits ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F ). Hence, its base = 16
- The symbols A, B, C, D, E and F represent the decimal values 10, 11, 12, 13, 14 and 15 respectively
- The maximum value of a single digit 15 ( one less than the value of the base )
HEXADECIMAL NUMBER SYSTEMS
- Each position of a digit represents a specific power of the base ( 16 )
- Since there are only 16 digits , 4 bits ( 24 = 16 ) are sufficient to represent any hexadecimal number in binary
Example
1AF16 = ( 1 x 162 ) + ( A x 161 ) + ( F x 160 )
= 1 x 256 + 10 x 16 + 15 x 1
= 256 + 160 + 15
= 43110
Converting a number of another base a decimal number
- Step 1 : Determine the column ( positional ) value of each digit
- Multiply the obtained column values by the digits in the corresponding columns
- Calculate the sum of these products
Converting a number of another base a decimal number
Example:
40678 = ?10
40678 = 4 x 83 + 0 x 82 + 6 x 81 + 7 x 80
= 4 x 512 + 0 + 48 + 7
= 2048 + 0 + 48 + 7
= 210310